A linear code \(C\subseteq \mathbb {F}_q^n\)  is called cyclic if for every \(c=(c_0,c_1,\cdots ,c_{n-1})\in C\) , the vector \((c_{n-1},c_0,\cdots ,c_{n-2})\)  obtained by a cyclic shift of the coordinates of \(c\)  is also in \(C\) . It is well known that there is a one-to-one correspondence between cyclic codes of length \(n\)  over \(\mathbb {F}_q\)  and ideals of the ring \(\mathbb {F}_q [x]/\langle x^n-1\rangle \) , for example see [1]}. Under this correspondence, each cyclic code can be uniquely represented by a monic polynomial \(g(x)\) , where \(g(x)\)  is the minimal degree generator of the corresponding ideal. The polynomial \(g(x)\)  is called the generator polynomial of such cyclic code. Let \(\alpha \)  be a fixed
primitive \(n\) -th root of unity in \(\mathbb {F}_{q^r}\) ,
where \(r\)  is the smallest positive integer such that
\(n|(q^r-1)\) .
Alternatively, we can represent the above cyclic code by its unique defining set
\(\lbrace t: 0\le t \le n-1 \ \text{and}\ g(\alpha ^t)=0 \rbrace .\) 
